Question: We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$?
We proceed as follows: \begin{align*}
(x - 2)(2x + 5) &= 8x - 6\\
2x^2 + x - 10 &= 8x - 6\\
2x^2 - 7x - 4 &= 0\\
(x - 4)(2x + 1) &= 0.
\end{align*}This gives us $x = 4$ or $x = -\frac{1}{2}.$ However, since the latter would make $x - 2$ negative, we see that $x = \boxed{4}.$